Practice Maths

Unit 1 Topic 1 Review — Consumer Arithmetic — Solutions

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This review covers all five lessons in Consumer Arithmetic: percentages and applications, earning income and taxation, simple interest, compound interest, and comparing financial options. Allow approximately 60–75 minutes for this review.

Review Questions

  1. Q1 — GST calculations

    Fluency

    (a) An item costs $85.00 before GST. Find the price including 10% GST.

    (b) A television is marked $132.00 including GST. Find the pre-GST price.

    (a) Adding GST:

    Price including GST = $85.00 × 1.10 = $93.50

    (Or: GST amount = $85.00 × 0.10 = $8.50, total = $85.00 + $8.50 = $93.50)

    (b) Removing GST:

    Pre-GST price = $132.00 ÷ 1.10 = $120.00

    (The GST included = $132.00 − $120.00 = $12.00)

  2. Q2 — Percentage change

    Fluency

    (a) A house was bought for $420 000 and sold for $483 000. Find the percentage increase.

    (b) A television is marked down 15% from $1200. Find the sale price.

    (a) Percentage increase:

    Increase = $483 000 − $420 000 = $63 000

    % increase = (63 000 ÷ 420 000) × 100 = 15%

    (b) Sale price after 15% markdown:

    Discount = $1200 × 0.15 = $180

    Sale price = $1200 − $180 = $1020

    (Or: Sale price = $1200 × 0.85 = $1020)

  3. Q3 — Calculating wages with overtime

    Fluency

    Jasmine earns $24.60 per hour. She works 38 hours at the normal rate and 4 hours overtime at time-and-a-half. Calculate her weekly gross pay.

    Normal pay:

    38 × $24.60 = $934.80

    Overtime rate:

    Time-and-a-half = $24.60 × 1.5 = $36.90 per hour

    Overtime pay:

    4 × $36.90 = $147.60

    Weekly gross pay:

    $934.80 + $147.60 = $1082.40

  4. Q4 — Net income calculation

    Fluency

    Sam earns a salary of $68 000. His tax is $14 422 and superannuation is $6800.

    (a) Find total deductions.

    (b) Find net income.

    (c) Express net income as a percentage of gross income.

    (a) Total deductions:

    $14 422 + $6800 = $21 222

    (b) Net income:

    $68 000 − $21 222 = $46 778

    (c) Net income as % of gross:

    (46 778 ÷ 68 000) × 100 = 68.8% (to 1 d.p.)

  5. Q5 — Simple interest calculations

    Fluency

    $4500 is invested at 3.8% p.a. simple interest for 2.5 years.

    (a) Find the interest earned.

    (b) Find the total amount at the end of the investment.

    (c) What rate would produce $513 interest on the same principal over the same time?

    P = $4500, r = 0.038, t = 2.5 years

    (a) Interest earned:

    I = Prt = 4500 × 0.038 × 2.5 = $427.50

    (b) Total amount:

    A = P + I = $4500 + $427.50 = $4927.50

    (c) Finding the rate for I = $513:

    I = Prt ⇒ r = I ÷ (Pt)

    r = 513 ÷ (4500 × 2.5) = 513 ÷ 11250 = 0.0456 = 4.56% p.a.

  6. Q6 — Simple interest vs compound interest

    Understanding

    $12 000 is invested at 6% p.a. for 5 years. Compare the final amounts under simple interest and compound interest (compounded annually). How much more does compound interest earn?

    P = $12 000, r = 0.06, t = 5 years

    Simple interest:

    I = 12000 × 0.06 × 5 = $3600

    Final amount = $15 600

    Compound interest (annually):

    A = 12000 × (1.06)5

    (1.06)5 = 1.33823

    A = 12000 × 1.33823 = $16 058.71

    Difference: $16 058.71 − $15 600 = $458.71

    Compound interest earns $458.71 more over 5 years.

  7. Q7 — Compound interest with monthly compounding

    Understanding

    $8000 is invested at 5.4% p.a. compounding monthly for 3 years.

    (a) Find the final amount.

    (b) Find the interest earned.

    (c) Find the effective annual rate (EAR).

    P = $8000, r = 0.054, n = 12 (monthly), t = 3 years

    (a) Final amount:

    A = 8000 × (1 + 0.054/12)12×3

    = 8000 × (1.0045)36

    (1.0045)36 = 1.17484 (approx)

    A = 8000 × 1.17484 = $9398.72

    (b) Interest earned:

    I = $9398.72 − $8000 = $1398.72

    (c) Effective Annual Rate:

    EAR = (1 + 0.054/12)12 − 1

    = (1.0045)12 − 1

    = 1.05536 − 1 = 0.05536

    EAR = 5.536% (compared to nominal 5.4%)

  8. Q8 — Finding the principal for a target amount

    Understanding

    How much must be invested at 4% p.a. compounding quarterly to accumulate $25 000 in 6 years?

    A = $25 000, r = 0.04, n = 4 (quarterly), t = 6 years

    A = P(1 + r/n)nt

    25000 = P × (1 + 0.04/4)4×6

    25000 = P × (1.01)24

    (1.01)24 = 1.26973 (approx)

    P = 25000 ÷ 1.26973 = $19 689.00

    Approximately $19 689 must be invested today.

  9. Q9 — Comparing accounts using EAR

    Understanding

    Account A: 5.2% p.a. compounding annually. Account B: 5.0% p.a. compounding monthly.

    (a) Find the EAR for each account.

    (b) Which account is better?

    (c) For $15 000 invested over 4 years, how much more does the better account earn?

    (a) EAR for Account A:

    EAR = (1 + 0.052/1)1 − 1 = 5.200%

    EAR for Account B:

    EAR = (1 + 0.05/12)12 − 1

    = (1.004167)12 − 1 = 1.05116 − 1 = 5.116%

    (b) Account A is better (EAR 5.200% > 5.116%)

    (c) Final amounts for $15 000 over 4 years:

    Account A: A = 15000 × (1.052)4 = 15000 × 1.22749 = $18 412.35

    Account B: A = 15000 × (1.004167)48

    (1.004167)48 = 1.22079 (approx)

    A = 15000 × 1.22079 = $18 311.85

    Difference: $18 412.35 − $18 311.85 = $100.50

    Account A earns $100.50 more over 4 years.

  10. Q10 — Commission income

    Understanding

    James is a real estate agent who earns a base salary of $2000 per month plus 1.8% commission on all sales. In March he sold properties worth $1.2 million. Find his gross income for March.

    Base salary: $2000

    Commission:

    Commission = 1.8% × $1 200 000 = 0.018 × 1 200 000 = $21 600

    Gross income for March:

    $2000 + $21 600 = $23 600

  11. Q11 — Loan comparison: simple vs compound

    Understanding

    An $18 000 loan is available on two options:

    Option A: 9% p.a. simple interest for 4 years.

    Option B: 8.4% p.a. compounding monthly for 4 years.

    Find the total repayment for each option. Which is cheaper, and by how much?

    Option A: 9% p.a. simple interest, 4 years

    I = 18000 × 0.09 × 4 = $6480

    Total repayment = $18 000 + $6480 = $24 480

    Option B: 8.4% p.a. compounding monthly, 4 years

    r = 0.084/12 = 0.007, n = 48

    A = 18000 × (1.007)48

    (1.007)48 = 1.39392 (approx)

    A = 18000 × 1.39392 = $25 090.56

    Option A is cheaper by: $25 090.56 − $24 480 = $610.56

    Despite the higher nominal rate, Option A (simple interest) costs less over 4 years than Option B (monthly compounding).

  12. Q12 — Investment target: find the principal

    Problem Solving

    Sofia wants to have $40 000 in 8 years. She finds a term deposit paying 3.6% p.a. compounding quarterly. What principal must she invest today?

    A = $40 000, r = 0.036, n = 4, t = 8 years

    A = P(1 + r/n)nt

    40000 = P × (1 + 0.036/4)32

    40000 = P × (1.009)32

    (1.009)32: ln(1.009) = 0.008960, ×32 = 0.28671, e0.28671 = 1.33218

    40000 = P × 1.33218

    P = 40000 ÷ 1.33218 = $30 025.60

    Sofia must invest approximately $30 026 today.

  13. Q13 — Income tax calculation using tax brackets

    Problem Solving

    Tax brackets (simplified):

    • $0 – $18 200: Nil
    • $18 201 – $45 000: 19c per $1 over $18 200
    • $45 001 – $120 000: $5092 + 32.5c per $1 over $45 000

    Calculate the income tax payable on a taxable income of $72 000.

    Taxable income = $72 000 → falls in the $45 001–$120 000 bracket.

    Amount over $45 000 = $72 000 − $45 000 = $27 000

    Tax = $5092 + (0.325 × $27 000)

    Tax = $5092 + $8775

    Tax = $13 867

    Verification: effective tax rate = 13 867 ÷ 72 000 × 100 = 19.3%

  14. Q14 — Rule of 72 and doubling time

    Problem Solving

    Use the Rule of 72 to estimate the doubling time for investments earning: (a) 4% p.a., (b) 6% p.a., (c) 9% p.a.

    Then verify your answer for (b) using A = P(1.06)t to confirm when $5000 first exceeds $10 000.

    Rule of 72: Doubling time ≈ 72 ÷ rate

    (a) 4% p.a.: 72 ÷ 4 = 18 years

    (b) 6% p.a.: 72 ÷ 6 = 12 years

    (c) 9% p.a.: 72 ÷ 9 = 8 years

    Verification for (b): 6% p.a. compound

    We need A = P(1.06)t > 10000 where P = 5000

    5000 × (1.06)t = 10000 ⇒ (1.06)t = 2

    Check t = 11: (1.06)11 = 1.8983 → $9491 (not yet doubled)

    Check t = 12: (1.06)12 = 2.0122 → $10 061 (doubled!)

    $5000 first exceeds $10 000 after 12 years, confirming the Rule of 72 estimate.

  15. Q15 — Total career income with pay rises and overtime

    Problem Solving

    Maya works 40 hours per week at $32 per hour. After 2 years she gets a 5% pay rise; after another 2 years she gets another 5% rise. She also works 3 hours overtime at time-and-a-half every week throughout. Working 52 weeks per year, calculate her total income across all 4 years.

    Years 1–2: Base rate $32/hr

    Normal weekly pay: 40 × $32 = $1280

    Overtime rate: $32 × 1.5 = $48/hr

    Overtime weekly pay: 3 × $48 = $144

    Weekly total: $1280 + $144 = $1424

    Annual income (Yr 1): 52 × $1424 = $74 048

    Total for Years 1–2: 2 × $74 048 = $148 096

    Years 3–4: Rate after 5% rise = $32 × 1.05 = $33.60/hr

    Normal weekly pay: 40 × $33.60 = $1344

    Overtime rate: $33.60 × 1.5 = $50.40/hr

    Overtime weekly pay: 3 × $50.40 = $151.20

    Weekly total: $1344 + $151.20 = $1495.20

    Annual income (Yr 3): 52 × $1495.20 = $77 750.40

    Total for Years 3–4: 2 × $77 750.40 = $155 500.80

    Note: The second 5% rise at Year 5 would affect Year 5 onwards — the question states “after another 2 years” which is the start of Year 5, so Years 3–4 use the first-rise rate only.

    Total income across 4 years:

    $148 096 + $155 500.80 = $303 596.80